Calculations in the Newman–Penrose (NP) formalism of general relativity normally begin with the construction of a complex null tetrad
\{la,na,ma,\bar{m}a\}
\{la,na\}
\{ma,\bar{m}a\}
(-,+,+,+):
la
a=n | |
l | |
a |
a=m | |
n | |
a |
a=\bar{m} | |
m | |
a |
\bar{m}a=0;
la
a=l | |
m | |
a |
a=n | |
\bar{m} | |
a |
a=n | |
m | |
a |
\bar{m}a=0;
lana=lana=-1, ma\bar{m}a=ma\bar{m}a=1;
gab=-lanb-nalb+ma\bar{m}b+\bar{m}amb, gab=-lanb-nalb+ma\bar{m}b+\bar{m}amb.
Only after the tetrad
\{la,na,ma,\bar{m}a\}
\Psii
\Phiij
\phii
la
na
ma
\bar{m}a
In the context below, it will be shown how these three methods work.
Note: In addition to the convention
\{(-,+,+,+);la
a | |
n | |
a=-1,m |
\bar{m}a=1\}
\{(+,-,-,-);la
a | |
n | |
a=1,m |
\bar{m}a=-1\}
The primary method to construct a complex null tetrad is via combinations of orthonormal bases. For a spacetime
gab
\{\omega0,\omega1,\omega2,\omega3\}
gab=-\omega0\omega0+\omega1\omega1+\omega2\omega2+\omega3\omega3,
the covectors
\{la,na,ma,\bar{m}a\}
| ||||
l | ||||
adx |
| ||||
m | ||||
adx |
and the tetrad vectors
\{la,na,ma,\bar{m}a\}
\{la,na,ma,\bar{m}a\}
gab
Remark: The nonholonomic construction is actually in accordance with the local light cone structure.
Example: A nonholonomic tetrad
Given a spacetime metric of the form (in signature(-,+,+,+))
gab=-gtt
2+g | |
dt | |
rr |
2+g | |
dr | |
\theta\theta |
2+g | |
d\theta | |
\phi\phi |
d\phi2,
the nonholonomic orthonormal covectors are therefore
\omegat=\sqrt{gtt
and the nonholonomic null covectors are therefore
| ||||
l | ||||
adx |
| ||||
n | ||||
adx |
| ||||
m | ||||
adx |
| ||||
\bar{m} | ||||
adx |
In Minkowski spacetime, the nonholonomically constructed null vectors
\{la,na\}
\{la,na\}
\{t,r,\theta,\phi\}
\{u,r,\theta,\phi\}
\{v,r,\theta,\phi\}
u
v
Example: Null tetrad for Schwarzschild metric in Eddington-Finkelstein coordinates reads
ds2=-Fdv2+2dvdr+r2(d\theta2+\sin2\thetad\phi2), withF:=(1-
2M | |
r |
),
so the Lagrangian for null radial geodesics of the Schwarzschild spacetime is
L=-Fv |
| |||
r |
,
which has an ingoing solution
v |
=0
r | = |
F | |
2 |
v |
| ||||
l |
,0,0), na=(0,-1,0,0),
| ||||
m |
r}(0,0,1,i\csc\theta),
and the dual basis covectors are therefore
l | ||||
|
,1,0,0), na=(-1,0,0,0),
m | ||||
|
Here we utilized the cross-normalization condition
al | |
l | |
a=-1 |
gab+lanb+nalb
hAB
dv
dr
\kappa=\sigma=\tau=0, \nu=λ=\pi=0, \gamma=0
\rho= | -r+2M |
2r2 |
, \mu=-
1 | |
r |
, \alpha=-\beta=
-\sqrt{2 | \varepsilon= | |
\cot\theta}{4r}, |
M | |
2r2 |
;
\Psi0=\Psi1=\Psi3=\Psi4=0,
\Psi | ||||
|
,
\Phi00=\Phi10=\Phi20=\Phi11=\Phi12=\Phi22=Λ=0.
Example: Null tetrad for extremal Reissner–Nordström metric in Eddington-Finkelstein coordinates reads
ds2=-Gdv2+2dvdr+r2d\theta2+r2\sin2\thetad\phi2, withG:=(1-
M | |
r |
)2,
so the Lagrangian is
2L=-G
v |
| |||
r+r |
2({
\theta} |
2+\sin
| |||
2).
For null radial geodesics with
\{L=0,
|
v=0 |
|
and therefore the tetrad for an ingoing observer can be set up as
a\partial | ||
l | (1, | |
a= |
F | |
2 |
,0,0),
a\partial | |
n | |
a=(0,-1,0,0 |
),
a= | ||
l | (- | |
adx |
F | |
2 |
,1,0,0),
a=(-1,0,0,0 | |
n | |
adx |
),
a\partial | ||||
m | ||||
|
With the tetrad defined, we are now able to work out the spin coefficients, Weyl-NP scalars and Ricci-NP scalars that
\kappa=\sigma=\tau=0, \nu=λ=\pi=0, \gamma=0
\rho= | (r-M)2 |
2r3 |
, \mu=-
1 | |
r |
, \alpha=-\beta=
-\sqrt{2 | \varepsilon= | |
\cot\theta}{4r}, |
M(r-M) | |
2r3 |
;
\Psi0=\Psi1=\Psi3=\Psi4=0,
\Psi | ||||
|
,
\Phi00=\Phi10=\Phi20=\Phi12=\Phi22=Λ=0, \Phi11=-
M2 | |
2r4 |
.
At some typical boundary regions such as null infinity, timelike infinity, spacelike infinity, black hole horizons and cosmological horizons, null tetrads adapted to spacetime structures are usually employed to achieve the most succinct Newman–Penrose descriptions.
For null infinity, the classic Newman-Unti (NU) tetrad[3] [4] [5] is employed to study asymptotic behaviors at null infinity,
a\partial | |
l | |
a=\partial |
r:=D,
a\partial | |
n | |
a=\partial |
u+U\partialr+X\partial\varsigma+\bar{X}\partial\bar:=\Delta,
a\partial | |
m | |
a=\omega\partial |
3\partial | |
\varsigma |
4\partial | |
+\xi | |
\bar\varsigma |
:=\delta,
a\partial | |
\bar{m} | |
a=\bar{\omega}\partial |
3\partial | |
\bar\varsigma |
4\partial | |
+\bar{\xi} | |
\varsigma |
:=\bar\delta,
where
\{U,X,\omega,\xi3,\xi4\}
u
la=du
r
la
a\partial | |
(Dr=l | |
ar=1) |
na
\Delta
a\partial | |
u=n | |
a |
u=1
\{u,r,\varsigma,\bar{\varsigma}\}
\{u,r\}
\{\varsigma:=ei\phi\cot
\theta | |
2 |
,\bar{\varsigma}=e-i\phi\cot
\theta | |
2 |
\}
\{\theta,\phi\}
2 | |
\hat\Delta | |
u |
Also, for the NU tetrad, the basic gauge conditions are
\kappa=\pi=\varepsilon=0, \rho=\bar\rho, \tau=\bar\alpha+\beta.
For a more comprehensive view of black holes in quasilocal definitions, adapted tetrads which can be smoothly transited from the exterior to the near-horizon vicinity and to the horizons are required. For example, for isolated horizons describing black holes in equilibrium with their exteriors, such a tetrad and the related coordinates can be constructed this way.[6] [7] [8] [9] [10] [11] Choose the first real null covector
na
na=-dv,
v
a\partial | |
l | |
a |
Dv=1, \Deltav=\deltav=\bar\deltav=0.
r
na
a\partial | |
n | |
a |
r=-1 \Leftrightarrow
a\partial | |
n | |
a |
=-\partialr.
Now, the first real null tetrad vector
na
\{la,ma,\barma\}
la
\{la,na,ma,\barma\}
a\partial | |
n | |
a |
\{ma,\barma\}
\{y,z\}
Tetrads satisfying the above restrictions can be expressed in the general form that
a\partial | |
l | |
a=\partial |
v+U\partialr
4 | |
+X | |
y+X |
\partial:=D,
a\partial | |
n | |
a=-\partial |
r:=\Delta,
a\partial | |
m | |
a=\Omega\partial |
3\partial | |
y |
4\partial | |
+\xi | |
z |
:=\delta,
a\partial | |
\bar{m} | |
a=\bar{\Omega}\partial |
r
3\partial | |
+\bar{\xi} | |
y |
4\partial | |
+\bar{\xi} | |
z |
:=\bar\delta.
The gauge conditions in this tetrad are
\nu=\tau=\gamma=0, \mu=\bar\mu, \pi=\alpha+\bar\beta,
Remark: Unlike Schwarzschild-type coordinates, here r=0 represents the horizon, while r>0 (r<0) corresponds to the exterior (interior) of an isolated horizon. People often Taylor expand a scalar
Q
Q=\sumi=0Q(i)ri=Q(0)+Q(1)r+ … +Q(n)rn+\ldots
where
Q(0)